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For the above types of convergence the random variables have to be defined on the same probability space. There is a variant without this assumption. 2 [Convergence in distribution] Let (Ωn , Fn , Pn ) and (Ω, F, P) be probability spaces and let fn : Ωn → ❘ and f : Ω → ❘ be random variables. Then the sequence (fn )∞ n=1 converges in distribution d to f (fn → f ) if and only if ❊ψ(fn) → ψ(f ) as n → ∞ for all bounded and continuous functions ψ : ❘ → ❘. 63 64 CHAPTER 4. MODES OF CONVERGENCE We have the following relations between the above types of convergence.